(Please write your Roll No. immediately)

Roll No. ..................................

Time : 3 Hours

Maximum Marks : 75

Note : Attempt any 5 questions in all. At least two questions from each section. All questions carry equal marks.

Q1

( a )

State and prove Lagrange's Mean value theorem and verify it for the function f(x) = sin x + cos x in [0,π/2].

5

( b )

If y = (sin^-1 x)², prove that (1 – x²) y₂ - xy₁ = 2 and also show that
(1 – x²) y_n+2 – (2n + 1) xy_n+1 – n²y_n = 0

5

( c )

If ρ₁ and ρ₂ are the radii of curvature at the extremities of two conjugate diameters of the ellipse
x²/a² + y²/b² = 1, then prove that (ρ₁^2/3 + ρ₂^2/3)(ab)^2/3 = a² + b²

5

Q2

( a )

Use Taylor's theorem to evaluate √10 correct to four significant figures.

5

( b )

Evaluate:

5

( c )

Determine the asymptotes of 4x³ - 3xy² - y³ + 2x² - xy - y² - 1 = 0

5

Q3

( a )

Show that
(i)

(ii)

5

( b )

Test for the convergence of the following series : -

1. x² + (2² / 3.4) x⁴ + (2² .4² / 3.4.5.6) x⁶ + (2² .4² .6² / 3.4.5.6.7.8) x⁸ + .....

2. log(2/1) – log(3/2) + log(4/3) – log(5/4) +.....

5

( c )

The velocity v(km/min) of a moped, which starts from rest is given at fixed intervals of time. Estimate approximately the distance covered in 20 minutes by applying Simpson's 1/3 rule.

5

t

2

4

6

8

10

12

14

16

18

20

v

10

18

25

29

32

20

11

5

2

0

Q4

( a )

Determine the length of the loop of the curve 9y² = (x-3)(x-6)²

5

( b )

Find the area common to the cardiode r = a(1+cos θ) and the circle r = (3/2)a and also the area of the remainder of the cardiode.

5

( c )

If the curve r = a+b cos θ (a>b) revolves about the initial line, show that the volume generated is (4/3) π a (a² +b² )

5

Q5

( a )

If v = (x² +y² +z² )^-1/2 show that

5

( b )

If

prove that:
(i)

(ii)

5

( c )

Expand sin(xy) in powers of (x-1) and (y-π/2) up to and including second degree terms.

5

Q6

( a )

Find the volume of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1

5

( b )

Sketch the region of integration and evaluate:

5

( c )

Changing into polar coordinates and hence evaluate:

5

Q7

( a )

Solve:
( i )

( ii )

5

( b )

Solve : ( 2xy cos x² – 2xy +1 )dx + ( sin² - x² )dy = 0

5

( c )

Solve :

5

Q8

( a )

If u₁=(x₂ x₃)/x₁ , u₂=(x₁ x₃)/x₂ , u₃=(x₁ x₂)/x₃ , then evaluate J ( u₁, u₂, u₃).

5

( b )

Solve :

5

( c )

Solve :

5